/* * Multi-precision integer library * * Copyright The Mbed TLS Contributors * SPDX-License-Identifier: Apache-2.0 * * Licensed under the Apache License, Version 2.0 (the "License"); you may * not use this file except in compliance with the License. * You may obtain a copy of the License at * * http://www.apache.org/licenses/LICENSE-2.0 * * Unless required by applicable law or agreed to in writing, software * distributed under the License is distributed on an "AS IS" BASIS, WITHOUT * WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. * See the License for the specific language governing permissions and * limitations under the License. */ /* * The following sources were referenced in the design of this Multi-precision * Integer library: * * [1] Handbook of Applied Cryptography - 1997 * Menezes, van Oorschot and Vanstone * * [2] Multi-Precision Math * Tom St Denis * https://github.com/libtom/libtommath/blob/develop/tommath.pdf * * [3] GNU Multi-Precision Arithmetic Library * https://gmplib.org/manual/index.html * */ #include "common.h" #if defined(MBEDTLS_BIGNUM_C) #include "mbedtls/bignum.h" #include "mbedtls/bn_mul.h" #include "mbedtls/platform_util.h" #include "mbedtls/error.h" #include "constant_time_internal.h" #include #if defined(MBEDTLS_PLATFORM_C) #include "mbedtls/platform.h" #else #include #include #define mbedtls_printf printf #define mbedtls_calloc calloc #define mbedtls_free free #endif #define MPI_VALIDATE_RET( cond ) \ MBEDTLS_INTERNAL_VALIDATE_RET( cond, MBEDTLS_ERR_MPI_BAD_INPUT_DATA ) #define MPI_VALIDATE( cond ) \ MBEDTLS_INTERNAL_VALIDATE( cond ) #define ciL (sizeof(mbedtls_mpi_uint)) /* chars in limb */ #define biL (ciL << 3) /* bits in limb */ #define biH (ciL << 2) /* half limb size */ #define MPI_SIZE_T_MAX ( (size_t) -1 ) /* SIZE_T_MAX is not standard */ /* * Convert between bits/chars and number of limbs * Divide first in order to avoid potential overflows */ #define BITS_TO_LIMBS(i) ( (i) / biL + ( (i) % biL != 0 ) ) #define CHARS_TO_LIMBS(i) ( (i) / ciL + ( (i) % ciL != 0 ) ) /* Implementation that should never be optimized out by the compiler */ static void mbedtls_mpi_zeroize( mbedtls_mpi_uint *v, size_t n ) { mbedtls_platform_zeroize( v, ciL * n ); } /* * Initialize one MPI */ void mbedtls_mpi_init( mbedtls_mpi *X ) { MPI_VALIDATE( X != NULL ); X->s = 1; X->n = 0; X->p = NULL; } /* * Unallocate one MPI */ void mbedtls_mpi_free( mbedtls_mpi *X ) { if( X == NULL ) return; if( X->p != NULL ) { mbedtls_mpi_zeroize( X->p, X->n ); mbedtls_free( X->p ); } X->s = 1; X->n = 0; X->p = NULL; } /* * Enlarge to the specified number of limbs */ int mbedtls_mpi_grow( mbedtls_mpi *X, size_t nblimbs ) { mbedtls_mpi_uint *p; MPI_VALIDATE_RET( X != NULL ); if( nblimbs > MBEDTLS_MPI_MAX_LIMBS ) return( MBEDTLS_ERR_MPI_ALLOC_FAILED ); if( X->n < nblimbs ) { if( ( p = (mbedtls_mpi_uint*)mbedtls_calloc( nblimbs, ciL ) ) == NULL ) return( MBEDTLS_ERR_MPI_ALLOC_FAILED ); if( X->p != NULL ) { memcpy( p, X->p, X->n * ciL ); mbedtls_mpi_zeroize( X->p, X->n ); mbedtls_free( X->p ); } X->n = nblimbs; X->p = p; } return( 0 ); } /* * Resize down as much as possible, * while keeping at least the specified number of limbs */ int mbedtls_mpi_shrink( mbedtls_mpi *X, size_t nblimbs ) { mbedtls_mpi_uint *p; size_t i; MPI_VALIDATE_RET( X != NULL ); if( nblimbs > MBEDTLS_MPI_MAX_LIMBS ) return( MBEDTLS_ERR_MPI_ALLOC_FAILED ); /* Actually resize up if there are currently fewer than nblimbs limbs. */ if( X->n <= nblimbs ) return( mbedtls_mpi_grow( X, nblimbs ) ); /* After this point, then X->n > nblimbs and in particular X->n > 0. */ for( i = X->n - 1; i > 0; i-- ) if( X->p[i] != 0 ) break; i++; if( i < nblimbs ) i = nblimbs; if( ( p = (mbedtls_mpi_uint*)mbedtls_calloc( i, ciL ) ) == NULL ) return( MBEDTLS_ERR_MPI_ALLOC_FAILED ); if( X->p != NULL ) { memcpy( p, X->p, i * ciL ); mbedtls_mpi_zeroize( X->p, X->n ); mbedtls_free( X->p ); } X->n = i; X->p = p; return( 0 ); } /* Resize X to have exactly n limbs and set it to 0. */ static int mbedtls_mpi_resize_clear( mbedtls_mpi *X, size_t limbs ) { if( limbs == 0 ) { mbedtls_mpi_free( X ); return( 0 ); } else if( X->n == limbs ) { memset( X->p, 0, limbs * ciL ); X->s = 1; return( 0 ); } else { mbedtls_mpi_free( X ); return( mbedtls_mpi_grow( X, limbs ) ); } } /* * Copy the contents of Y into X. * * This function is not constant-time. Leading zeros in Y may be removed. * * Ensure that X does not shrink. This is not guaranteed by the public API, * but some code in the bignum module relies on this property, for example * in mbedtls_mpi_exp_mod(). */ int mbedtls_mpi_copy( mbedtls_mpi *X, const mbedtls_mpi *Y ) { int ret = 0; size_t i; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( Y != NULL ); if( X == Y ) return( 0 ); if( Y->n == 0 ) { if( X->n != 0 ) { X->s = 1; memset( X->p, 0, X->n * ciL ); } return( 0 ); } for( i = Y->n - 1; i > 0; i-- ) if( Y->p[i] != 0 ) break; i++; X->s = Y->s; if( X->n < i ) { MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, i ) ); } else { memset( X->p + i, 0, ( X->n - i ) * ciL ); } memcpy( X->p, Y->p, i * ciL ); cleanup: return( ret ); } /* * Swap the contents of X and Y */ void mbedtls_mpi_swap( mbedtls_mpi *X, mbedtls_mpi *Y ) { mbedtls_mpi T; MPI_VALIDATE( X != NULL ); MPI_VALIDATE( Y != NULL ); memcpy( &T, X, sizeof( mbedtls_mpi ) ); memcpy( X, Y, sizeof( mbedtls_mpi ) ); memcpy( Y, &T, sizeof( mbedtls_mpi ) ); } /* * Set value from integer */ int mbedtls_mpi_lset( mbedtls_mpi *X, mbedtls_mpi_sint z ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MPI_VALIDATE_RET( X != NULL ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, 1 ) ); memset( X->p, 0, X->n * ciL ); X->p[0] = ( z < 0 ) ? -z : z; X->s = ( z < 0 ) ? -1 : 1; cleanup: return( ret ); } /* * Get a specific bit */ int mbedtls_mpi_get_bit( const mbedtls_mpi *X, size_t pos ) { MPI_VALIDATE_RET( X != NULL ); if( X->n * biL <= pos ) return( 0 ); return( ( X->p[pos / biL] >> ( pos % biL ) ) & 0x01 ); } /* Get a specific byte, without range checks. */ #define GET_BYTE( X, i ) \ ( ( ( X )->p[( i ) / ciL] >> ( ( ( i ) % ciL ) * 8 ) ) & 0xff ) /* * Set a bit to a specific value of 0 or 1 */ int mbedtls_mpi_set_bit( mbedtls_mpi *X, size_t pos, unsigned char val ) { int ret = 0; size_t off = pos / biL; size_t idx = pos % biL; MPI_VALIDATE_RET( X != NULL ); if( val != 0 && val != 1 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); if( X->n * biL <= pos ) { if( val == 0 ) return( 0 ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, off + 1 ) ); } X->p[off] &= ~( (mbedtls_mpi_uint) 0x01 << idx ); X->p[off] |= (mbedtls_mpi_uint) val << idx; cleanup: return( ret ); } /* * Return the number of less significant zero-bits */ size_t mbedtls_mpi_lsb( const mbedtls_mpi *X ) { size_t i, j, count = 0; MBEDTLS_INTERNAL_VALIDATE_RET( X != NULL, 0 ); for( i = 0; i < X->n; i++ ) for( j = 0; j < biL; j++, count++ ) if( ( ( X->p[i] >> j ) & 1 ) != 0 ) return( count ); return( 0 ); } /* * Count leading zero bits in a given integer */ static size_t mbedtls_clz( const mbedtls_mpi_uint x ) { size_t j; mbedtls_mpi_uint mask = (mbedtls_mpi_uint) 1 << (biL - 1); for( j = 0; j < biL; j++ ) { if( x & mask ) break; mask >>= 1; } return j; } /* * Return the number of bits */ size_t mbedtls_mpi_bitlen( const mbedtls_mpi *X ) { size_t i, j; if( X->n == 0 ) return( 0 ); for( i = X->n - 1; i > 0; i-- ) if( X->p[i] != 0 ) break; j = biL - mbedtls_clz( X->p[i] ); return( ( i * biL ) + j ); } /* * Return the total size in bytes */ size_t mbedtls_mpi_size( const mbedtls_mpi *X ) { return( ( mbedtls_mpi_bitlen( X ) + 7 ) >> 3 ); } /* * Convert an ASCII character to digit value */ static int mpi_get_digit( mbedtls_mpi_uint *d, int radix, char c ) { *d = 255; if( c >= 0x30 && c <= 0x39 ) *d = c - 0x30; if( c >= 0x41 && c <= 0x46 ) *d = c - 0x37; if( c >= 0x61 && c <= 0x66 ) *d = c - 0x57; if( *d >= (mbedtls_mpi_uint) radix ) return( MBEDTLS_ERR_MPI_INVALID_CHARACTER ); return( 0 ); } /* * Import from an ASCII string */ int mbedtls_mpi_read_string( mbedtls_mpi *X, int radix, const char *s ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, j, slen, n; int sign = 1; mbedtls_mpi_uint d; mbedtls_mpi T; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( s != NULL ); if( radix < 2 || radix > 16 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); mbedtls_mpi_init( &T ); if( s[0] == 0 ) { mbedtls_mpi_free( X ); return( 0 ); } if( s[0] == '-' ) { ++s; sign = -1; } slen = strlen( s ); if( radix == 16 ) { if( slen > MPI_SIZE_T_MAX >> 2 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); n = BITS_TO_LIMBS( slen << 2 ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, n ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( X, 0 ) ); for( i = slen, j = 0; i > 0; i--, j++ ) { MBEDTLS_MPI_CHK( mpi_get_digit( &d, radix, s[i - 1] ) ); X->p[j / ( 2 * ciL )] |= d << ( ( j % ( 2 * ciL ) ) << 2 ); } } else { MBEDTLS_MPI_CHK( mbedtls_mpi_lset( X, 0 ) ); for( i = 0; i < slen; i++ ) { MBEDTLS_MPI_CHK( mpi_get_digit( &d, radix, s[i] ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &T, X, radix ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( X, &T, d ) ); } } if( sign < 0 && mbedtls_mpi_bitlen( X ) != 0 ) X->s = -1; cleanup: mbedtls_mpi_free( &T ); return( ret ); } /* * Helper to write the digits high-order first. */ static int mpi_write_hlp( mbedtls_mpi *X, int radix, char **p, const size_t buflen ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi_uint r; size_t length = 0; char *p_end = *p + buflen; do { if( length >= buflen ) { return( MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL ); } MBEDTLS_MPI_CHK( mbedtls_mpi_mod_int( &r, X, radix ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_div_int( X, NULL, X, radix ) ); /* * Write the residue in the current position, as an ASCII character. */ if( r < 0xA ) *(--p_end) = (char)( '0' + r ); else *(--p_end) = (char)( 'A' + ( r - 0xA ) ); length++; } while( mbedtls_mpi_cmp_int( X, 0 ) != 0 ); memmove( *p, p_end, length ); *p += length; cleanup: return( ret ); } /* * Export into an ASCII string */ int mbedtls_mpi_write_string( const mbedtls_mpi *X, int radix, char *buf, size_t buflen, size_t *olen ) { int ret = 0; size_t n; char *p; mbedtls_mpi T; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( olen != NULL ); MPI_VALIDATE_RET( buflen == 0 || buf != NULL ); if( radix < 2 || radix > 16 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); n = mbedtls_mpi_bitlen( X ); /* Number of bits necessary to present `n`. */ if( radix >= 4 ) n >>= 1; /* Number of 4-adic digits necessary to present * `n`. If radix > 4, this might be a strict * overapproximation of the number of * radix-adic digits needed to present `n`. */ if( radix >= 16 ) n >>= 1; /* Number of hexadecimal digits necessary to * present `n`. */ n += 1; /* Terminating null byte */ n += 1; /* Compensate for the divisions above, which round down `n` * in case it's not even. */ n += 1; /* Potential '-'-sign. */ n += ( n & 1 ); /* Make n even to have enough space for hexadecimal writing, * which always uses an even number of hex-digits. */ if( buflen < n ) { *olen = n; return( MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL ); } p = buf; mbedtls_mpi_init( &T ); if( X->s == -1 ) { *p++ = '-'; buflen--; } if( radix == 16 ) { int c; size_t i, j, k; for( i = X->n, k = 0; i > 0; i-- ) { for( j = ciL; j > 0; j-- ) { c = ( X->p[i - 1] >> ( ( j - 1 ) << 3) ) & 0xFF; if( c == 0 && k == 0 && ( i + j ) != 2 ) continue; *(p++) = "0123456789ABCDEF" [c / 16]; *(p++) = "0123456789ABCDEF" [c % 16]; k = 1; } } } else { MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &T, X ) ); if( T.s == -1 ) T.s = 1; MBEDTLS_MPI_CHK( mpi_write_hlp( &T, radix, &p, buflen ) ); } *p++ = '\0'; *olen = p - buf; cleanup: mbedtls_mpi_free( &T ); return( ret ); } #if defined(MBEDTLS_FS_IO) /* * Read X from an opened file */ int mbedtls_mpi_read_file( mbedtls_mpi *X, int radix, FILE *fin ) { mbedtls_mpi_uint d; size_t slen; char *p; /* * Buffer should have space for (short) label and decimal formatted MPI, * newline characters and '\0' */ char s[ MBEDTLS_MPI_RW_BUFFER_SIZE ]; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( fin != NULL ); if( radix < 2 || radix > 16 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); memset( s, 0, sizeof( s ) ); if( fgets( s, sizeof( s ) - 1, fin ) == NULL ) return( MBEDTLS_ERR_MPI_FILE_IO_ERROR ); slen = strlen( s ); if( slen == sizeof( s ) - 2 ) return( MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL ); if( slen > 0 && s[slen - 1] == '\n' ) { slen--; s[slen] = '\0'; } if( slen > 0 && s[slen - 1] == '\r' ) { slen--; s[slen] = '\0'; } p = s + slen; while( p-- > s ) if( mpi_get_digit( &d, radix, *p ) != 0 ) break; return( mbedtls_mpi_read_string( X, radix, p + 1 ) ); } /* * Write X into an opened file (or stdout if fout == NULL) */ int mbedtls_mpi_write_file( const char *p, const mbedtls_mpi *X, int radix, FILE *fout ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t n, slen, plen; /* * Buffer should have space for (short) label and decimal formatted MPI, * newline characters and '\0' */ char s[ MBEDTLS_MPI_RW_BUFFER_SIZE ]; MPI_VALIDATE_RET( X != NULL ); if( radix < 2 || radix > 16 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); memset( s, 0, sizeof( s ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_write_string( X, radix, s, sizeof( s ) - 2, &n ) ); if( p == NULL ) p = ""; plen = strlen( p ); slen = strlen( s ); s[slen++] = '\r'; s[slen++] = '\n'; if( fout != NULL ) { if( fwrite( p, 1, plen, fout ) != plen || fwrite( s, 1, slen, fout ) != slen ) return( MBEDTLS_ERR_MPI_FILE_IO_ERROR ); } else mbedtls_printf( "%s%s", p, s ); cleanup: return( ret ); } #endif /* MBEDTLS_FS_IO */ /* Convert a big-endian byte array aligned to the size of mbedtls_mpi_uint * into the storage form used by mbedtls_mpi. */ static mbedtls_mpi_uint mpi_uint_bigendian_to_host_c( mbedtls_mpi_uint x ) { uint8_t i; unsigned char *x_ptr; mbedtls_mpi_uint tmp = 0; for( i = 0, x_ptr = (unsigned char*) &x; i < ciL; i++, x_ptr++ ) { tmp <<= CHAR_BIT; tmp |= (mbedtls_mpi_uint) *x_ptr; } return( tmp ); } static mbedtls_mpi_uint mpi_uint_bigendian_to_host( mbedtls_mpi_uint x ) { #if defined(__BYTE_ORDER__) /* Nothing to do on bigendian systems. */ #if ( __BYTE_ORDER__ == __ORDER_BIG_ENDIAN__ ) return( x ); #endif /* __BYTE_ORDER__ == __ORDER_BIG_ENDIAN__ */ #if ( __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__ ) /* For GCC and Clang, have builtins for byte swapping. */ #if defined(__GNUC__) && defined(__GNUC_PREREQ) #if __GNUC_PREREQ(4,3) #define have_bswap #endif #endif #if defined(__clang__) && defined(__has_builtin) #if __has_builtin(__builtin_bswap32) && \ __has_builtin(__builtin_bswap64) #define have_bswap #endif #endif #if defined(have_bswap) /* The compiler is hopefully able to statically evaluate this! */ switch( sizeof(mbedtls_mpi_uint) ) { case 4: return( __builtin_bswap32(x) ); case 8: return( __builtin_bswap64(x) ); } #endif #endif /* __BYTE_ORDER__ == __ORDER_LITTLE_ENDIAN__ */ #endif /* __BYTE_ORDER__ */ /* Fall back to C-based reordering if we don't know the byte order * or we couldn't use a compiler-specific builtin. */ return( mpi_uint_bigendian_to_host_c( x ) ); } static void mpi_bigendian_to_host( mbedtls_mpi_uint * const p, size_t limbs ) { mbedtls_mpi_uint *cur_limb_left; mbedtls_mpi_uint *cur_limb_right; if( limbs == 0 ) return; /* * Traverse limbs and * - adapt byte-order in each limb * - swap the limbs themselves. * For that, simultaneously traverse the limbs from left to right * and from right to left, as long as the left index is not bigger * than the right index (it's not a problem if limbs is odd and the * indices coincide in the last iteration). */ for( cur_limb_left = p, cur_limb_right = p + ( limbs - 1 ); cur_limb_left <= cur_limb_right; cur_limb_left++, cur_limb_right-- ) { mbedtls_mpi_uint tmp; /* Note that if cur_limb_left == cur_limb_right, * this code effectively swaps the bytes only once. */ tmp = mpi_uint_bigendian_to_host( *cur_limb_left ); *cur_limb_left = mpi_uint_bigendian_to_host( *cur_limb_right ); *cur_limb_right = tmp; } } /* * Import X from unsigned binary data, little endian */ int mbedtls_mpi_read_binary_le( mbedtls_mpi *X, const unsigned char *buf, size_t buflen ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i; size_t const limbs = CHARS_TO_LIMBS( buflen ); /* Ensure that target MPI has exactly the necessary number of limbs */ MBEDTLS_MPI_CHK( mbedtls_mpi_resize_clear( X, limbs ) ); for( i = 0; i < buflen; i++ ) X->p[i / ciL] |= ((mbedtls_mpi_uint) buf[i]) << ((i % ciL) << 3); cleanup: /* * This function is also used to import keys. However, wiping the buffers * upon failure is not necessary because failure only can happen before any * input is copied. */ return( ret ); } /* * Import X from unsigned binary data, big endian */ int mbedtls_mpi_read_binary( mbedtls_mpi *X, const unsigned char *buf, size_t buflen ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t const limbs = CHARS_TO_LIMBS( buflen ); size_t const overhead = ( limbs * ciL ) - buflen; unsigned char *Xp; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( buflen == 0 || buf != NULL ); /* Ensure that target MPI has exactly the necessary number of limbs */ MBEDTLS_MPI_CHK( mbedtls_mpi_resize_clear( X, limbs ) ); /* Avoid calling `memcpy` with NULL source or destination argument, * even if buflen is 0. */ if( buflen != 0 ) { Xp = (unsigned char*) X->p; memcpy( Xp + overhead, buf, buflen ); mpi_bigendian_to_host( X->p, limbs ); } cleanup: /* * This function is also used to import keys. However, wiping the buffers * upon failure is not necessary because failure only can happen before any * input is copied. */ return( ret ); } /* * Export X into unsigned binary data, little endian */ int mbedtls_mpi_write_binary_le( const mbedtls_mpi *X, unsigned char *buf, size_t buflen ) { size_t stored_bytes = X->n * ciL; size_t bytes_to_copy; size_t i; if( stored_bytes < buflen ) { bytes_to_copy = stored_bytes; } else { bytes_to_copy = buflen; /* The output buffer is smaller than the allocated size of X. * However X may fit if its leading bytes are zero. */ for( i = bytes_to_copy; i < stored_bytes; i++ ) { if( GET_BYTE( X, i ) != 0 ) return( MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL ); } } for( i = 0; i < bytes_to_copy; i++ ) buf[i] = GET_BYTE( X, i ); if( stored_bytes < buflen ) { /* Write trailing 0 bytes */ memset( buf + stored_bytes, 0, buflen - stored_bytes ); } return( 0 ); } /* * Export X into unsigned binary data, big endian */ int mbedtls_mpi_write_binary( const mbedtls_mpi *X, unsigned char *buf, size_t buflen ) { size_t stored_bytes; size_t bytes_to_copy; unsigned char *p; size_t i; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( buflen == 0 || buf != NULL ); stored_bytes = X->n * ciL; if( stored_bytes < buflen ) { /* There is enough space in the output buffer. Write initial * null bytes and record the position at which to start * writing the significant bytes. In this case, the execution * trace of this function does not depend on the value of the * number. */ bytes_to_copy = stored_bytes; p = buf + buflen - stored_bytes; memset( buf, 0, buflen - stored_bytes ); } else { /* The output buffer is smaller than the allocated size of X. * However X may fit if its leading bytes are zero. */ bytes_to_copy = buflen; p = buf; for( i = bytes_to_copy; i < stored_bytes; i++ ) { if( GET_BYTE( X, i ) != 0 ) return( MBEDTLS_ERR_MPI_BUFFER_TOO_SMALL ); } } for( i = 0; i < bytes_to_copy; i++ ) p[bytes_to_copy - i - 1] = GET_BYTE( X, i ); return( 0 ); } /* * Left-shift: X <<= count */ int mbedtls_mpi_shift_l( mbedtls_mpi *X, size_t count ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, v0, t1; mbedtls_mpi_uint r0 = 0, r1; MPI_VALIDATE_RET( X != NULL ); v0 = count / (biL ); t1 = count & (biL - 1); i = mbedtls_mpi_bitlen( X ) + count; if( X->n * biL < i ) MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, BITS_TO_LIMBS( i ) ) ); ret = 0; /* * shift by count / limb_size */ if( v0 > 0 ) { for( i = X->n; i > v0; i-- ) X->p[i - 1] = X->p[i - v0 - 1]; for( ; i > 0; i-- ) X->p[i - 1] = 0; } /* * shift by count % limb_size */ if( t1 > 0 ) { for( i = v0; i < X->n; i++ ) { r1 = X->p[i] >> (biL - t1); X->p[i] <<= t1; X->p[i] |= r0; r0 = r1; } } cleanup: return( ret ); } /* * Right-shift: X >>= count */ int mbedtls_mpi_shift_r( mbedtls_mpi *X, size_t count ) { size_t i, v0, v1; mbedtls_mpi_uint r0 = 0, r1; MPI_VALIDATE_RET( X != NULL ); v0 = count / biL; v1 = count & (biL - 1); if( v0 > X->n || ( v0 == X->n && v1 > 0 ) ) return mbedtls_mpi_lset( X, 0 ); /* * shift by count / limb_size */ if( v0 > 0 ) { for( i = 0; i < X->n - v0; i++ ) X->p[i] = X->p[i + v0]; for( ; i < X->n; i++ ) X->p[i] = 0; } /* * shift by count % limb_size */ if( v1 > 0 ) { for( i = X->n; i > 0; i-- ) { r1 = X->p[i - 1] << (biL - v1); X->p[i - 1] >>= v1; X->p[i - 1] |= r0; r0 = r1; } } return( 0 ); } /* * Compare unsigned values */ int mbedtls_mpi_cmp_abs( const mbedtls_mpi *X, const mbedtls_mpi *Y ) { size_t i, j; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( Y != NULL ); for( i = X->n; i > 0; i-- ) if( X->p[i - 1] != 0 ) break; for( j = Y->n; j > 0; j-- ) if( Y->p[j - 1] != 0 ) break; if( i == 0 && j == 0 ) return( 0 ); if( i > j ) return( 1 ); if( j > i ) return( -1 ); for( ; i > 0; i-- ) { if( X->p[i - 1] > Y->p[i - 1] ) return( 1 ); if( X->p[i - 1] < Y->p[i - 1] ) return( -1 ); } return( 0 ); } /* * Compare signed values */ int mbedtls_mpi_cmp_mpi( const mbedtls_mpi *X, const mbedtls_mpi *Y ) { size_t i, j; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( Y != NULL ); for( i = X->n; i > 0; i-- ) if( X->p[i - 1] != 0 ) break; for( j = Y->n; j > 0; j-- ) if( Y->p[j - 1] != 0 ) break; if( i == 0 && j == 0 ) return( 0 ); if( i > j ) return( X->s ); if( j > i ) return( -Y->s ); if( X->s > 0 && Y->s < 0 ) return( 1 ); if( Y->s > 0 && X->s < 0 ) return( -1 ); for( ; i > 0; i-- ) { if( X->p[i - 1] > Y->p[i - 1] ) return( X->s ); if( X->p[i - 1] < Y->p[i - 1] ) return( -X->s ); } return( 0 ); } /* * Compare signed values */ int mbedtls_mpi_cmp_int( const mbedtls_mpi *X, mbedtls_mpi_sint z ) { mbedtls_mpi Y; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET( X != NULL ); *p = ( z < 0 ) ? -z : z; Y.s = ( z < 0 ) ? -1 : 1; Y.n = 1; Y.p = p; return( mbedtls_mpi_cmp_mpi( X, &Y ) ); } /* * Unsigned addition: X = |A| + |B| (HAC 14.7) */ int mbedtls_mpi_add_abs( mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, j; mbedtls_mpi_uint *o, *p, c, tmp; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); if( X == B ) { const mbedtls_mpi *T = A; A = X; B = T; } if( X != A ) MBEDTLS_MPI_CHK( mbedtls_mpi_copy( X, A ) ); /* * X should always be positive as a result of unsigned additions. */ X->s = 1; for( j = B->n; j > 0; j-- ) if( B->p[j - 1] != 0 ) break; MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, j ) ); o = B->p; p = X->p; c = 0; /* * tmp is used because it might happen that p == o */ for( i = 0; i < j; i++, o++, p++ ) { tmp= *o; *p += c; c = ( *p < c ); *p += tmp; c += ( *p < tmp ); } while( c != 0 ) { if( i >= X->n ) { MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, i + 1 ) ); p = X->p + i; } *p += c; c = ( *p < c ); i++; p++; } cleanup: return( ret ); } /** * Helper for mbedtls_mpi subtraction. * * Calculate l - r where l and r have the same size. * This function operates modulo (2^ciL)^n and returns the carry * (1 if there was a wraparound, i.e. if `l < r`, and 0 otherwise). * * d may be aliased to l or r. * * \param n Number of limbs of \p d, \p l and \p r. * \param[out] d The result of the subtraction. * \param[in] l The left operand. * \param[in] r The right operand. * * \return 1 if `l < r`. * 0 if `l >= r`. */ static mbedtls_mpi_uint mpi_sub_hlp( size_t n, mbedtls_mpi_uint *d, const mbedtls_mpi_uint *l, const mbedtls_mpi_uint *r ) { size_t i; mbedtls_mpi_uint c = 0, t, z; for( i = 0; i < n; i++ ) { z = ( l[i] < c ); t = l[i] - c; c = ( t < r[i] ) + z; d[i] = t - r[i]; } return( c ); } /* * Unsigned subtraction: X = |A| - |B| (HAC 14.9, 14.10) */ int mbedtls_mpi_sub_abs( mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t n; mbedtls_mpi_uint carry; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); for( n = B->n; n > 0; n-- ) if( B->p[n - 1] != 0 ) break; if( n > A->n ) { /* B >= (2^ciL)^n > A */ ret = MBEDTLS_ERR_MPI_NEGATIVE_VALUE; goto cleanup; } MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, A->n ) ); /* Set the high limbs of X to match A. Don't touch the lower limbs * because X might be aliased to B, and we must not overwrite the * significant digits of B. */ if( A->n > n ) memcpy( X->p + n, A->p + n, ( A->n - n ) * ciL ); if( X->n > A->n ) memset( X->p + A->n, 0, ( X->n - A->n ) * ciL ); carry = mpi_sub_hlp( n, X->p, A->p, B->p ); if( carry != 0 ) { /* Propagate the carry to the first nonzero limb of X. */ for( ; n < X->n && X->p[n] == 0; n++ ) --X->p[n]; /* If we ran out of space for the carry, it means that the result * is negative. */ if( n == X->n ) { ret = MBEDTLS_ERR_MPI_NEGATIVE_VALUE; goto cleanup; } --X->p[n]; } /* X should always be positive as a result of unsigned subtractions. */ X->s = 1; cleanup: return( ret ); } /* * Signed addition: X = A + B */ int mbedtls_mpi_add_mpi( mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret, s; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); s = A->s; if( A->s * B->s < 0 ) { if( mbedtls_mpi_cmp_abs( A, B ) >= 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( X, A, B ) ); X->s = s; } else { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( X, B, A ) ); X->s = -s; } } else { MBEDTLS_MPI_CHK( mbedtls_mpi_add_abs( X, A, B ) ); X->s = s; } cleanup: return( ret ); } /* * Signed subtraction: X = A - B */ int mbedtls_mpi_sub_mpi( mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret, s; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); s = A->s; if( A->s * B->s > 0 ) { if( mbedtls_mpi_cmp_abs( A, B ) >= 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( X, A, B ) ); X->s = s; } else { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( X, B, A ) ); X->s = -s; } } else { MBEDTLS_MPI_CHK( mbedtls_mpi_add_abs( X, A, B ) ); X->s = s; } cleanup: return( ret ); } /* * Signed addition: X = A + b */ int mbedtls_mpi_add_int( mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_sint b ) { mbedtls_mpi B; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); p[0] = ( b < 0 ) ? -b : b; B.s = ( b < 0 ) ? -1 : 1; B.n = 1; B.p = p; return( mbedtls_mpi_add_mpi( X, A, &B ) ); } /* * Signed subtraction: X = A - b */ int mbedtls_mpi_sub_int( mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_sint b ) { mbedtls_mpi B; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); p[0] = ( b < 0 ) ? -b : b; B.s = ( b < 0 ) ? -1 : 1; B.n = 1; B.p = p; return( mbedtls_mpi_sub_mpi( X, A, &B ) ); } /** Helper for mbedtls_mpi multiplication. * * Add \p b * \p s to \p d. * * \param i The number of limbs of \p s. * \param[in] s A bignum to multiply, of size \p i. * It may overlap with \p d, but only if * \p d <= \p s. * Its leading limb must not be \c 0. * \param[in,out] d The bignum to add to. * It must be sufficiently large to store the * result of the multiplication. This means * \p i + 1 limbs if \p d[\p i - 1] started as 0 and \p b * is not known a priori. * \param b A scalar to multiply. */ static #if defined(__APPLE__) && defined(__arm__) /* * Apple LLVM version 4.2 (clang-425.0.24) (based on LLVM 3.2svn) * appears to need this to prevent bad ARM code generation at -O3. */ __attribute__ ((noinline)) #endif void mpi_mul_hlp( size_t i, const mbedtls_mpi_uint *s, mbedtls_mpi_uint *d, mbedtls_mpi_uint b ) { mbedtls_mpi_uint c = 0, t = 0; #if defined(MULADDC_HUIT) for( ; i >= 8; i -= 8 ) { MULADDC_INIT MULADDC_HUIT MULADDC_STOP } for( ; i > 0; i-- ) { MULADDC_INIT MULADDC_CORE MULADDC_STOP } #else /* MULADDC_HUIT */ for( ; i >= 16; i -= 16 ) { MULADDC_INIT MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_STOP } for( ; i >= 8; i -= 8 ) { MULADDC_INIT MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_CORE MULADDC_STOP } for( ; i > 0; i-- ) { MULADDC_INIT MULADDC_CORE MULADDC_STOP } #endif /* MULADDC_HUIT */ t++; while( c != 0 ) { *d += c; c = ( *d < c ); d++; } } /* * Baseline multiplication: X = A * B (HAC 14.12) */ int mbedtls_mpi_mul_mpi( mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, j; mbedtls_mpi TA, TB; int result_is_zero = 0; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); mbedtls_mpi_init( &TA ); mbedtls_mpi_init( &TB ); if( X == A ) { MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &TA, A ) ); A = &TA; } if( X == B ) { MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &TB, B ) ); B = &TB; } for( i = A->n; i > 0; i-- ) if( A->p[i - 1] != 0 ) break; if( i == 0 ) result_is_zero = 1; for( j = B->n; j > 0; j-- ) if( B->p[j - 1] != 0 ) break; if( j == 0 ) result_is_zero = 1; MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, i + j ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( X, 0 ) ); for( ; j > 0; j-- ) mpi_mul_hlp( i, A->p, X->p + j - 1, B->p[j - 1] ); /* If the result is 0, we don't shortcut the operation, which reduces * but does not eliminate side channels leaking the zero-ness. We do * need to take care to set the sign bit properly since the library does * not fully support an MPI object with a value of 0 and s == -1. */ if( result_is_zero ) X->s = 1; else X->s = A->s * B->s; cleanup: mbedtls_mpi_free( &TB ); mbedtls_mpi_free( &TA ); return( ret ); } /* * Baseline multiplication: X = A * b */ int mbedtls_mpi_mul_int( mbedtls_mpi *X, const mbedtls_mpi *A, mbedtls_mpi_uint b ) { MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); /* mpi_mul_hlp can't deal with a leading 0. */ size_t n = A->n; while( n > 0 && A->p[n - 1] == 0 ) --n; /* The general method below doesn't work if n==0 or b==0. By chance * calculating the result is trivial in those cases. */ if( b == 0 || n == 0 ) { return( mbedtls_mpi_lset( X, 0 ) ); } /* Calculate A*b as A + A*(b-1) to take advantage of mpi_mul_hlp */ int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; /* In general, A * b requires 1 limb more than b. If * A->p[n - 1] * b / b == A->p[n - 1], then A * b fits in the same * number of limbs as A and the call to grow() is not required since * copy() will take care of the growth if needed. However, experimentally, * making the call to grow() unconditional causes slightly fewer * calls to calloc() in ECP code, presumably because it reuses the * same mpi for a while and this way the mpi is more likely to directly * grow to its final size. */ MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, n + 1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( X, A ) ); mpi_mul_hlp( n, A->p, X->p, b - 1 ); cleanup: return( ret ); } /* * Unsigned integer divide - double mbedtls_mpi_uint dividend, u1/u0, and * mbedtls_mpi_uint divisor, d */ static mbedtls_mpi_uint mbedtls_int_div_int( mbedtls_mpi_uint u1, mbedtls_mpi_uint u0, mbedtls_mpi_uint d, mbedtls_mpi_uint *r ) { #if defined(MBEDTLS_HAVE_UDBL) mbedtls_t_udbl dividend, quotient; #else const mbedtls_mpi_uint radix = (mbedtls_mpi_uint) 1 << biH; const mbedtls_mpi_uint uint_halfword_mask = ( (mbedtls_mpi_uint) 1 << biH ) - 1; mbedtls_mpi_uint d0, d1, q0, q1, rAX, r0, quotient; mbedtls_mpi_uint u0_msw, u0_lsw; size_t s; #endif /* * Check for overflow */ if( 0 == d || u1 >= d ) { if (r != NULL) *r = ~0; return ( ~0 ); } #if defined(MBEDTLS_HAVE_UDBL) dividend = (mbedtls_t_udbl) u1 << biL; dividend |= (mbedtls_t_udbl) u0; quotient = dividend / d; if( quotient > ( (mbedtls_t_udbl) 1 << biL ) - 1 ) quotient = ( (mbedtls_t_udbl) 1 << biL ) - 1; if( r != NULL ) *r = (mbedtls_mpi_uint)( dividend - (quotient * d ) ); return (mbedtls_mpi_uint) quotient; #else /* * Algorithm D, Section 4.3.1 - The Art of Computer Programming * Vol. 2 - Seminumerical Algorithms, Knuth */ /* * Normalize the divisor, d, and dividend, u0, u1 */ s = mbedtls_clz( d ); d = d << s; u1 = u1 << s; u1 |= ( u0 >> ( biL - s ) ) & ( -(mbedtls_mpi_sint)s >> ( biL - 1 ) ); u0 = u0 << s; d1 = d >> biH; d0 = d & uint_halfword_mask; u0_msw = u0 >> biH; u0_lsw = u0 & uint_halfword_mask; /* * Find the first quotient and remainder */ q1 = u1 / d1; r0 = u1 - d1 * q1; while( q1 >= radix || ( q1 * d0 > radix * r0 + u0_msw ) ) { q1 -= 1; r0 += d1; if ( r0 >= radix ) break; } rAX = ( u1 * radix ) + ( u0_msw - q1 * d ); q0 = rAX / d1; r0 = rAX - q0 * d1; while( q0 >= radix || ( q0 * d0 > radix * r0 + u0_lsw ) ) { q0 -= 1; r0 += d1; if ( r0 >= radix ) break; } if (r != NULL) *r = ( rAX * radix + u0_lsw - q0 * d ) >> s; quotient = q1 * radix + q0; return quotient; #endif } /* * Division by mbedtls_mpi: A = Q * B + R (HAC 14.20) */ int mbedtls_mpi_div_mpi( mbedtls_mpi *Q, mbedtls_mpi *R, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t i, n, t, k; mbedtls_mpi X, Y, Z, T1, T2; mbedtls_mpi_uint TP2[3]; MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); if( mbedtls_mpi_cmp_int( B, 0 ) == 0 ) return( MBEDTLS_ERR_MPI_DIVISION_BY_ZERO ); mbedtls_mpi_init( &X ); mbedtls_mpi_init( &Y ); mbedtls_mpi_init( &Z ); mbedtls_mpi_init( &T1 ); /* * Avoid dynamic memory allocations for constant-size T2. * * T2 is used for comparison only and the 3 limbs are assigned explicitly, * so nobody increase the size of the MPI and we're safe to use an on-stack * buffer. */ T2.s = 1; T2.n = sizeof( TP2 ) / sizeof( *TP2 ); T2.p = TP2; if( mbedtls_mpi_cmp_abs( A, B ) < 0 ) { if( Q != NULL ) MBEDTLS_MPI_CHK( mbedtls_mpi_lset( Q, 0 ) ); if( R != NULL ) MBEDTLS_MPI_CHK( mbedtls_mpi_copy( R, A ) ); return( 0 ); } MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &X, A ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &Y, B ) ); X.s = Y.s = 1; MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &Z, A->n + 2 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &Z, 0 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &T1, A->n + 2 ) ); k = mbedtls_mpi_bitlen( &Y ) % biL; if( k < biL - 1 ) { k = biL - 1 - k; MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &X, k ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &Y, k ) ); } else k = 0; n = X.n - 1; t = Y.n - 1; MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &Y, biL * ( n - t ) ) ); while( mbedtls_mpi_cmp_mpi( &X, &Y ) >= 0 ) { Z.p[n - t]++; MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X, &X, &Y ) ); } MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &Y, biL * ( n - t ) ) ); for( i = n; i > t ; i-- ) { if( X.p[i] >= Y.p[t] ) Z.p[i - t - 1] = ~0; else { Z.p[i - t - 1] = mbedtls_int_div_int( X.p[i], X.p[i - 1], Y.p[t], NULL); } T2.p[0] = ( i < 2 ) ? 0 : X.p[i - 2]; T2.p[1] = ( i < 1 ) ? 0 : X.p[i - 1]; T2.p[2] = X.p[i]; Z.p[i - t - 1]++; do { Z.p[i - t - 1]--; MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &T1, 0 ) ); T1.p[0] = ( t < 1 ) ? 0 : Y.p[t - 1]; T1.p[1] = Y.p[t]; MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &T1, &T1, Z.p[i - t - 1] ) ); } while( mbedtls_mpi_cmp_mpi( &T1, &T2 ) > 0 ); MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &T1, &Y, Z.p[i - t - 1] ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &T1, biL * ( i - t - 1 ) ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X, &X, &T1 ) ); if( mbedtls_mpi_cmp_int( &X, 0 ) < 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &T1, &Y ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &T1, biL * ( i - t - 1 ) ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &X, &X, &T1 ) ); Z.p[i - t - 1]--; } } if( Q != NULL ) { MBEDTLS_MPI_CHK( mbedtls_mpi_copy( Q, &Z ) ); Q->s = A->s * B->s; } if( R != NULL ) { MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &X, k ) ); X.s = A->s; MBEDTLS_MPI_CHK( mbedtls_mpi_copy( R, &X ) ); if( mbedtls_mpi_cmp_int( R, 0 ) == 0 ) R->s = 1; } cleanup: mbedtls_mpi_free( &X ); mbedtls_mpi_free( &Y ); mbedtls_mpi_free( &Z ); mbedtls_mpi_free( &T1 ); mbedtls_platform_zeroize( TP2, sizeof( TP2 ) ); return( ret ); } /* * Division by int: A = Q * b + R */ int mbedtls_mpi_div_int( mbedtls_mpi *Q, mbedtls_mpi *R, const mbedtls_mpi *A, mbedtls_mpi_sint b ) { mbedtls_mpi B; mbedtls_mpi_uint p[1]; MPI_VALIDATE_RET( A != NULL ); p[0] = ( b < 0 ) ? -b : b; B.s = ( b < 0 ) ? -1 : 1; B.n = 1; B.p = p; return( mbedtls_mpi_div_mpi( Q, R, A, &B ) ); } /* * Modulo: R = A mod B */ int mbedtls_mpi_mod_mpi( mbedtls_mpi *R, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; MPI_VALIDATE_RET( R != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); if( mbedtls_mpi_cmp_int( B, 0 ) < 0 ) return( MBEDTLS_ERR_MPI_NEGATIVE_VALUE ); MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( NULL, R, A, B ) ); while( mbedtls_mpi_cmp_int( R, 0 ) < 0 ) MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( R, R, B ) ); while( mbedtls_mpi_cmp_mpi( R, B ) >= 0 ) MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( R, R, B ) ); cleanup: return( ret ); } /* * Modulo: r = A mod b */ int mbedtls_mpi_mod_int( mbedtls_mpi_uint *r, const mbedtls_mpi *A, mbedtls_mpi_sint b ) { size_t i; mbedtls_mpi_uint x, y, z; MPI_VALIDATE_RET( r != NULL ); MPI_VALIDATE_RET( A != NULL ); if( b == 0 ) return( MBEDTLS_ERR_MPI_DIVISION_BY_ZERO ); if( b < 0 ) return( MBEDTLS_ERR_MPI_NEGATIVE_VALUE ); /* * handle trivial cases */ if( b == 1 ) { *r = 0; return( 0 ); } if( b == 2 ) { *r = A->p[0] & 1; return( 0 ); } /* * general case */ for( i = A->n, y = 0; i > 0; i-- ) { x = A->p[i - 1]; y = ( y << biH ) | ( x >> biH ); z = y / b; y -= z * b; x <<= biH; y = ( y << biH ) | ( x >> biH ); z = y / b; y -= z * b; } /* * If A is negative, then the current y represents a negative value. * Flipping it to the positive side. */ if( A->s < 0 && y != 0 ) y = b - y; *r = y; return( 0 ); } /* * Fast Montgomery initialization (thanks to Tom St Denis) */ static void mpi_montg_init( mbedtls_mpi_uint *mm, const mbedtls_mpi *N ) { mbedtls_mpi_uint x, m0 = N->p[0]; unsigned int i; x = m0; x += ( ( m0 + 2 ) & 4 ) << 1; for( i = biL; i >= 8; i /= 2 ) x *= ( 2 - ( m0 * x ) ); *mm = ~x + 1; } /** Montgomery multiplication: A = A * B * R^-1 mod N (HAC 14.36) * * \param[in,out] A One of the numbers to multiply. * It must have at least as many limbs as N * (A->n >= N->n), and any limbs beyond n are ignored. * On successful completion, A contains the result of * the multiplication A * B * R^-1 mod N where * R = (2^ciL)^n. * \param[in] B One of the numbers to multiply. * It must be nonzero and must not have more limbs than N * (B->n <= N->n). * \param[in] N The modulo. N must be odd. * \param mm The value calculated by `mpi_montg_init(&mm, N)`. * This is -N^-1 mod 2^ciL. * \param[in,out] T A bignum for temporary storage. * It must be at least twice the limb size of N plus 2 * (T->n >= 2 * (N->n + 1)). * Its initial content is unused and * its final content is indeterminate. * Note that unlike the usual convention in the library * for `const mbedtls_mpi*`, the content of T can change. */ static void mpi_montmul( mbedtls_mpi *A, const mbedtls_mpi *B, const mbedtls_mpi *N, mbedtls_mpi_uint mm, const mbedtls_mpi *T ) { size_t i, n, m; mbedtls_mpi_uint u0, u1, *d; memset( T->p, 0, T->n * ciL ); d = T->p; n = N->n; m = ( B->n < n ) ? B->n : n; for( i = 0; i < n; i++ ) { /* * T = (T + u0*B + u1*N) / 2^biL */ u0 = A->p[i]; u1 = ( d[0] + u0 * B->p[0] ) * mm; mpi_mul_hlp( m, B->p, d, u0 ); mpi_mul_hlp( n, N->p, d, u1 ); *d++ = u0; d[n + 1] = 0; } /* At this point, d is either the desired result or the desired result * plus N. We now potentially subtract N, avoiding leaking whether the * subtraction is performed through side channels. */ /* Copy the n least significant limbs of d to A, so that * A = d if d < N (recall that N has n limbs). */ memcpy( A->p, d, n * ciL ); /* If d >= N then we want to set A to d - N. To prevent timing attacks, * do the calculation without using conditional tests. */ /* Set d to d0 + (2^biL)^n - N where d0 is the current value of d. */ d[n] += 1; d[n] -= mpi_sub_hlp( n, d, d, N->p ); /* If d0 < N then d < (2^biL)^n * so d[n] == 0 and we want to keep A as it is. * If d0 >= N then d >= (2^biL)^n, and d <= (2^biL)^n + N < 2 * (2^biL)^n * so d[n] == 1 and we want to set A to the result of the subtraction * which is d - (2^biL)^n, i.e. the n least significant limbs of d. * This exactly corresponds to a conditional assignment. */ mbedtls_ct_mpi_uint_cond_assign( n, A->p, d, (unsigned char) d[n] ); } /* * Montgomery reduction: A = A * R^-1 mod N * * See mpi_montmul() regarding constraints and guarantees on the parameters. */ static void mpi_montred( mbedtls_mpi *A, const mbedtls_mpi *N, mbedtls_mpi_uint mm, const mbedtls_mpi *T ) { mbedtls_mpi_uint z = 1; mbedtls_mpi U; U.n = U.s = (int) z; U.p = &z; mpi_montmul( A, &U, N, mm, T ); } /** * Select an MPI from a table without leaking the index. * * This is functionally equivalent to mbedtls_mpi_copy(R, T[idx]) except it * reads the entire table in order to avoid leaking the value of idx to an * attacker able to observe memory access patterns. * * \param[out] R Where to write the selected MPI. * \param[in] T The table to read from. * \param[in] T_size The number of elements in the table. * \param[in] idx The index of the element to select; * this must satisfy 0 <= idx < T_size. * * \return \c 0 on success, or a negative error code. */ static int mpi_select( mbedtls_mpi *R, const mbedtls_mpi *T, size_t T_size, size_t idx ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; for( size_t i = 0; i < T_size; i++ ) { MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( R, &T[i], (unsigned char) mbedtls_ct_size_bool_eq( i, idx ) ) ); } cleanup: return( ret ); } /* * Sliding-window exponentiation: X = A^E mod N (HAC 14.85) */ int mbedtls_mpi_exp_mod( mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *E, const mbedtls_mpi *N, mbedtls_mpi *prec_RR ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t wbits, wsize, one = 1; size_t i, j, nblimbs; size_t bufsize, nbits; mbedtls_mpi_uint ei, mm, state; mbedtls_mpi RR, T, W[ 1 << MBEDTLS_MPI_WINDOW_SIZE ], WW, Apos; int neg; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( E != NULL ); MPI_VALIDATE_RET( N != NULL ); if( mbedtls_mpi_cmp_int( N, 0 ) <= 0 || ( N->p[0] & 1 ) == 0 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); if( mbedtls_mpi_cmp_int( E, 0 ) < 0 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); if( mbedtls_mpi_bitlen( E ) > MBEDTLS_MPI_MAX_BITS || mbedtls_mpi_bitlen( N ) > MBEDTLS_MPI_MAX_BITS ) return ( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); /* * Init temps and window size */ mpi_montg_init( &mm, N ); mbedtls_mpi_init( &RR ); mbedtls_mpi_init( &T ); mbedtls_mpi_init( &Apos ); mbedtls_mpi_init( &WW ); memset( W, 0, sizeof( W ) ); i = mbedtls_mpi_bitlen( E ); wsize = ( i > 671 ) ? 6 : ( i > 239 ) ? 5 : ( i > 79 ) ? 4 : ( i > 23 ) ? 3 : 1; #if( MBEDTLS_MPI_WINDOW_SIZE < 6 ) if( wsize > MBEDTLS_MPI_WINDOW_SIZE ) wsize = MBEDTLS_MPI_WINDOW_SIZE; #endif j = N->n + 1; /* All W[i] and X must have at least N->n limbs for the mpi_montmul() * and mpi_montred() calls later. Here we ensure that W[1] and X are * large enough, and later we'll grow other W[i] to the same length. * They must not be shrunk midway through this function! */ MBEDTLS_MPI_CHK( mbedtls_mpi_grow( X, j ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &W[1], j ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &T, j * 2 ) ); /* * Compensate for negative A (and correct at the end) */ neg = ( A->s == -1 ); if( neg ) { MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &Apos, A ) ); Apos.s = 1; A = &Apos; } /* * If 1st call, pre-compute R^2 mod N */ if( prec_RR == NULL || prec_RR->p == NULL ) { MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &RR, 1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &RR, N->n * 2 * biL ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &RR, &RR, N ) ); if( prec_RR != NULL ) memcpy( prec_RR, &RR, sizeof( mbedtls_mpi ) ); } else memcpy( &RR, prec_RR, sizeof( mbedtls_mpi ) ); /* * W[1] = A * R^2 * R^-1 mod N = A * R mod N */ if( mbedtls_mpi_cmp_mpi( A, N ) >= 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &W[1], A, N ) ); /* This should be a no-op because W[1] is already that large before * mbedtls_mpi_mod_mpi(), but it's necessary to avoid an overflow * in mpi_montmul() below, so let's make sure. */ MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &W[1], N->n + 1 ) ); } else MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &W[1], A ) ); /* Note that this is safe because W[1] always has at least N->n limbs * (it grew above and was preserved by mbedtls_mpi_copy()). */ mpi_montmul( &W[1], &RR, N, mm, &T ); /* * X = R^2 * R^-1 mod N = R mod N */ MBEDTLS_MPI_CHK( mbedtls_mpi_copy( X, &RR ) ); mpi_montred( X, N, mm, &T ); if( wsize > 1 ) { /* * W[1 << (wsize - 1)] = W[1] ^ (wsize - 1) */ j = one << ( wsize - 1 ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &W[j], N->n + 1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &W[j], &W[1] ) ); for( i = 0; i < wsize - 1; i++ ) mpi_montmul( &W[j], &W[j], N, mm, &T ); /* * W[i] = W[i - 1] * W[1] */ for( i = j + 1; i < ( one << wsize ); i++ ) { MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &W[i], N->n + 1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &W[i], &W[i - 1] ) ); mpi_montmul( &W[i], &W[1], N, mm, &T ); } } nblimbs = E->n; bufsize = 0; nbits = 0; wbits = 0; state = 0; while( 1 ) { if( bufsize == 0 ) { if( nblimbs == 0 ) break; nblimbs--; bufsize = sizeof( mbedtls_mpi_uint ) << 3; } bufsize--; ei = (E->p[nblimbs] >> bufsize) & 1; /* * skip leading 0s */ if( ei == 0 && state == 0 ) continue; if( ei == 0 && state == 1 ) { /* * out of window, square X */ mpi_montmul( X, X, N, mm, &T ); continue; } /* * add ei to current window */ state = 2; nbits++; wbits |= ( ei << ( wsize - nbits ) ); if( nbits == wsize ) { /* * X = X^wsize R^-1 mod N */ for( i = 0; i < wsize; i++ ) mpi_montmul( X, X, N, mm, &T ); /* * X = X * W[wbits] R^-1 mod N */ MBEDTLS_MPI_CHK( mpi_select( &WW, W, (size_t) 1 << wsize, wbits ) ); mpi_montmul( X, &WW, N, mm, &T ); state--; nbits = 0; wbits = 0; } } /* * process the remaining bits */ for( i = 0; i < nbits; i++ ) { mpi_montmul( X, X, N, mm, &T ); wbits <<= 1; if( ( wbits & ( one << wsize ) ) != 0 ) mpi_montmul( X, &W[1], N, mm, &T ); } /* * X = A^E * R * R^-1 mod N = A^E mod N */ mpi_montred( X, N, mm, &T ); if( neg && E->n != 0 && ( E->p[0] & 1 ) != 0 ) { X->s = -1; MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( X, N, X ) ); } cleanup: for( i = ( one << ( wsize - 1 ) ); i < ( one << wsize ); i++ ) mbedtls_mpi_free( &W[i] ); mbedtls_mpi_free( &W[1] ); mbedtls_mpi_free( &T ); mbedtls_mpi_free( &Apos ); mbedtls_mpi_free( &WW ); if( prec_RR == NULL || prec_RR->p == NULL ) mbedtls_mpi_free( &RR ); return( ret ); } /* * Greatest common divisor: G = gcd(A, B) (HAC 14.54) */ int mbedtls_mpi_gcd( mbedtls_mpi *G, const mbedtls_mpi *A, const mbedtls_mpi *B ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t lz, lzt; mbedtls_mpi TA, TB; MPI_VALIDATE_RET( G != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( B != NULL ); mbedtls_mpi_init( &TA ); mbedtls_mpi_init( &TB ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &TA, A ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &TB, B ) ); lz = mbedtls_mpi_lsb( &TA ); lzt = mbedtls_mpi_lsb( &TB ); /* The loop below gives the correct result when A==0 but not when B==0. * So have a special case for B==0. Leverage the fact that we just * calculated the lsb and lsb(B)==0 iff B is odd or 0 to make the test * slightly more efficient than cmp_int(). */ if( lzt == 0 && mbedtls_mpi_get_bit( &TB, 0 ) == 0 ) { ret = mbedtls_mpi_copy( G, A ); goto cleanup; } if( lzt < lz ) lz = lzt; TA.s = TB.s = 1; /* We mostly follow the procedure described in HAC 14.54, but with some * minor differences: * - Sequences of multiplications or divisions by 2 are grouped into a * single shift operation. * - The procedure in HAC assumes that 0 < TB <= TA. * - The condition TB <= TA is not actually necessary for correctness. * TA and TB have symmetric roles except for the loop termination * condition, and the shifts at the beginning of the loop body * remove any significance from the ordering of TA vs TB before * the shifts. * - If TA = 0, the loop goes through 0 iterations and the result is * correctly TB. * - The case TB = 0 was short-circuited above. * * For the correctness proof below, decompose the original values of * A and B as * A = sa * 2^a * A' with A'=0 or A' odd, and sa = +-1 * B = sb * 2^b * B' with B'=0 or B' odd, and sb = +-1 * Then gcd(A, B) = 2^{min(a,b)} * gcd(A',B'), * and gcd(A',B') is odd or 0. * * At the beginning, we have TA = |A| and TB = |B| so gcd(A,B) = gcd(TA,TB). * The code maintains the following invariant: * gcd(A,B) = 2^k * gcd(TA,TB) for some k (I) */ /* Proof that the loop terminates: * At each iteration, either the right-shift by 1 is made on a nonzero * value and the nonnegative integer bitlen(TA) + bitlen(TB) decreases * by at least 1, or the right-shift by 1 is made on zero and then * TA becomes 0 which ends the loop (TB cannot be 0 if it is right-shifted * since in that case TB is calculated from TB-TA with the condition TB>TA). */ while( mbedtls_mpi_cmp_int( &TA, 0 ) != 0 ) { /* Divisions by 2 preserve the invariant (I). */ MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &TA, mbedtls_mpi_lsb( &TA ) ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &TB, mbedtls_mpi_lsb( &TB ) ) ); /* Set either TA or TB to |TA-TB|/2. Since TA and TB are both odd, * TA-TB is even so the division by 2 has an integer result. * Invariant (I) is preserved since any odd divisor of both TA and TB * also divides |TA-TB|/2, and any odd divisor of both TA and |TA-TB|/2 * also divides TB, and any odd divisior of both TB and |TA-TB|/2 also * divides TA. */ if( mbedtls_mpi_cmp_mpi( &TA, &TB ) >= 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( &TA, &TA, &TB ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &TA, 1 ) ); } else { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( &TB, &TB, &TA ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &TB, 1 ) ); } /* Note that one of TA or TB is still odd. */ } /* By invariant (I), gcd(A,B) = 2^k * gcd(TA,TB) for some k. * At the loop exit, TA = 0, so gcd(TA,TB) = TB. * - If there was at least one loop iteration, then one of TA or TB is odd, * and TA = 0, so TB is odd and gcd(TA,TB) = gcd(A',B'). In this case, * lz = min(a,b) so gcd(A,B) = 2^lz * TB. * - If there was no loop iteration, then A was 0, and gcd(A,B) = B. * In this case, lz = 0 and B = TB so gcd(A,B) = B = 2^lz * TB as well. */ MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &TB, lz ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( G, &TB ) ); cleanup: mbedtls_mpi_free( &TA ); mbedtls_mpi_free( &TB ); return( ret ); } /* Fill X with n_bytes random bytes. * X must already have room for those bytes. * The ordering of the bytes returned from the RNG is suitable for * deterministic ECDSA (see RFC 6979 §3.3 and mbedtls_mpi_random()). * The size and sign of X are unchanged. * n_bytes must not be 0. */ static int mpi_fill_random_internal( mbedtls_mpi *X, size_t n_bytes, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; const size_t limbs = CHARS_TO_LIMBS( n_bytes ); const size_t overhead = ( limbs * ciL ) - n_bytes; if( X->n < limbs ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); memset( X->p, 0, overhead ); memset( (unsigned char *) X->p + limbs * ciL, 0, ( X->n - limbs ) * ciL ); MBEDTLS_MPI_CHK( f_rng( p_rng, (unsigned char *) X->p + overhead, n_bytes ) ); mpi_bigendian_to_host( X->p, limbs ); cleanup: return( ret ); } /* * Fill X with size bytes of random. * * Use a temporary bytes representation to make sure the result is the same * regardless of the platform endianness (useful when f_rng is actually * deterministic, eg for tests). */ int mbedtls_mpi_fill_random( mbedtls_mpi *X, size_t size, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; size_t const limbs = CHARS_TO_LIMBS( size ); MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( f_rng != NULL ); /* Ensure that target MPI has exactly the necessary number of limbs */ MBEDTLS_MPI_CHK( mbedtls_mpi_resize_clear( X, limbs ) ); if( size == 0 ) return( 0 ); ret = mpi_fill_random_internal( X, size, f_rng, p_rng ); cleanup: return( ret ); } int mbedtls_mpi_random( mbedtls_mpi *X, mbedtls_mpi_sint min, const mbedtls_mpi *N, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret = MBEDTLS_ERR_MPI_BAD_INPUT_DATA; int count; unsigned lt_lower = 1, lt_upper = 0; size_t n_bits = mbedtls_mpi_bitlen( N ); size_t n_bytes = ( n_bits + 7 ) / 8; mbedtls_mpi lower_bound; if( min < 0 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); if( mbedtls_mpi_cmp_int( N, min ) <= 0 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); /* * When min == 0, each try has at worst a probability 1/2 of failing * (the msb has a probability 1/2 of being 0, and then the result will * be < N), so after 30 tries failure probability is a most 2**(-30). * * When N is just below a power of 2, as is the case when generating * a random scalar on most elliptic curves, 1 try is enough with * overwhelming probability. When N is just above a power of 2, * as when generating a random scalar on secp224k1, each try has * a probability of failing that is almost 1/2. * * The probabilities are almost the same if min is nonzero but negligible * compared to N. This is always the case when N is crypto-sized, but * it's convenient to support small N for testing purposes. When N * is small, use a higher repeat count, otherwise the probability of * failure is macroscopic. */ count = ( n_bytes > 4 ? 30 : 250 ); mbedtls_mpi_init( &lower_bound ); /* Ensure that target MPI has exactly the same number of limbs * as the upper bound, even if the upper bound has leading zeros. * This is necessary for the mbedtls_mpi_lt_mpi_ct() check. */ MBEDTLS_MPI_CHK( mbedtls_mpi_resize_clear( X, N->n ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_grow( &lower_bound, N->n ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &lower_bound, min ) ); /* * Match the procedure given in RFC 6979 §3.3 (deterministic ECDSA) * when f_rng is a suitably parametrized instance of HMAC_DRBG: * - use the same byte ordering; * - keep the leftmost n_bits bits of the generated octet string; * - try until result is in the desired range. * This also avoids any bias, which is especially important for ECDSA. */ do { MBEDTLS_MPI_CHK( mpi_fill_random_internal( X, n_bytes, f_rng, p_rng ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( X, 8 * n_bytes - n_bits ) ); if( --count == 0 ) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; goto cleanup; } MBEDTLS_MPI_CHK( mbedtls_mpi_lt_mpi_ct( X, &lower_bound, <_lower ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lt_mpi_ct( X, N, <_upper ) ); } while( lt_lower != 0 || lt_upper == 0 ); cleanup: mbedtls_mpi_free( &lower_bound ); return( ret ); } /* * Modular inverse: X = A^-1 mod N (HAC 14.61 / 14.64) */ int mbedtls_mpi_inv_mod( mbedtls_mpi *X, const mbedtls_mpi *A, const mbedtls_mpi *N ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi G, TA, TU, U1, U2, TB, TV, V1, V2; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( A != NULL ); MPI_VALIDATE_RET( N != NULL ); if( mbedtls_mpi_cmp_int( N, 1 ) <= 0 ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); mbedtls_mpi_init( &TA ); mbedtls_mpi_init( &TU ); mbedtls_mpi_init( &U1 ); mbedtls_mpi_init( &U2 ); mbedtls_mpi_init( &G ); mbedtls_mpi_init( &TB ); mbedtls_mpi_init( &TV ); mbedtls_mpi_init( &V1 ); mbedtls_mpi_init( &V2 ); MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( &G, A, N ) ); if( mbedtls_mpi_cmp_int( &G, 1 ) != 0 ) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; goto cleanup; } MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &TA, A, N ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &TU, &TA ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &TB, N ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &TV, N ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &U1, 1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &U2, 0 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &V1, 0 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &V2, 1 ) ); do { while( ( TU.p[0] & 1 ) == 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &TU, 1 ) ); if( ( U1.p[0] & 1 ) != 0 || ( U2.p[0] & 1 ) != 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &U1, &U1, &TB ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &U2, &U2, &TA ) ); } MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &U1, 1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &U2, 1 ) ); } while( ( TV.p[0] & 1 ) == 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &TV, 1 ) ); if( ( V1.p[0] & 1 ) != 0 || ( V2.p[0] & 1 ) != 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &V1, &V1, &TB ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &V2, &V2, &TA ) ); } MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &V1, 1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &V2, 1 ) ); } if( mbedtls_mpi_cmp_mpi( &TU, &TV ) >= 0 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &TU, &TU, &TV ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &U1, &U1, &V1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &U2, &U2, &V2 ) ); } else { MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &TV, &TV, &TU ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &V1, &V1, &U1 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &V2, &V2, &U2 ) ); } } while( mbedtls_mpi_cmp_int( &TU, 0 ) != 0 ); while( mbedtls_mpi_cmp_int( &V1, 0 ) < 0 ) MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &V1, &V1, N ) ); while( mbedtls_mpi_cmp_mpi( &V1, N ) >= 0 ) MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &V1, &V1, N ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( X, &V1 ) ); cleanup: mbedtls_mpi_free( &TA ); mbedtls_mpi_free( &TU ); mbedtls_mpi_free( &U1 ); mbedtls_mpi_free( &U2 ); mbedtls_mpi_free( &G ); mbedtls_mpi_free( &TB ); mbedtls_mpi_free( &TV ); mbedtls_mpi_free( &V1 ); mbedtls_mpi_free( &V2 ); return( ret ); } #if defined(MBEDTLS_GENPRIME) static const int small_prime[] = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, -103 }; /* * Small divisors test (X must be positive) * * Return values: * 0: no small factor (possible prime, more tests needed) * 1: certain prime * MBEDTLS_ERR_MPI_NOT_ACCEPTABLE: certain non-prime * other negative: error */ static int mpi_check_small_factors( const mbedtls_mpi *X ) { int ret = 0; size_t i; mbedtls_mpi_uint r; if( ( X->p[0] & 1 ) == 0 ) return( MBEDTLS_ERR_MPI_NOT_ACCEPTABLE ); for( i = 0; small_prime[i] > 0; i++ ) { if( mbedtls_mpi_cmp_int( X, small_prime[i] ) <= 0 ) return( 1 ); MBEDTLS_MPI_CHK( mbedtls_mpi_mod_int( &r, X, small_prime[i] ) ); if( r == 0 ) return( MBEDTLS_ERR_MPI_NOT_ACCEPTABLE ); } cleanup: return( ret ); } /* * Miller-Rabin pseudo-primality test (HAC 4.24) */ static int mpi_miller_rabin( const mbedtls_mpi *X, size_t rounds, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret, count; size_t i, j, k, s; mbedtls_mpi W, R, T, A, RR; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( f_rng != NULL ); mbedtls_mpi_init( &W ); mbedtls_mpi_init( &R ); mbedtls_mpi_init( &T ); mbedtls_mpi_init( &A ); mbedtls_mpi_init( &RR ); /* * W = |X| - 1 * R = W >> lsb( W ) */ MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &W, X, 1 ) ); s = mbedtls_mpi_lsb( &W ); MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R, &W ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &R, s ) ); for( i = 0; i < rounds; i++ ) { /* * pick a random A, 1 < A < |X| - 1 */ count = 0; do { MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &A, X->n * ciL, f_rng, p_rng ) ); j = mbedtls_mpi_bitlen( &A ); k = mbedtls_mpi_bitlen( &W ); if (j > k) { A.p[A.n - 1] &= ( (mbedtls_mpi_uint) 1 << ( k - ( A.n - 1 ) * biL - 1 ) ) - 1; } if (count++ > 30) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; goto cleanup; } } while ( mbedtls_mpi_cmp_mpi( &A, &W ) >= 0 || mbedtls_mpi_cmp_int( &A, 1 ) <= 0 ); /* * A = A^R mod |X| */ MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &A, &A, &R, X, &RR ) ); if( mbedtls_mpi_cmp_mpi( &A, &W ) == 0 || mbedtls_mpi_cmp_int( &A, 1 ) == 0 ) continue; j = 1; while( j < s && mbedtls_mpi_cmp_mpi( &A, &W ) != 0 ) { /* * A = A * A mod |X| */ MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, &A, &A ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_mod_mpi( &A, &T, X ) ); if( mbedtls_mpi_cmp_int( &A, 1 ) == 0 ) break; j++; } /* * not prime if A != |X| - 1 or A == 1 */ if( mbedtls_mpi_cmp_mpi( &A, &W ) != 0 || mbedtls_mpi_cmp_int( &A, 1 ) == 0 ) { ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; break; } } cleanup: mbedtls_mpi_free( &W ); mbedtls_mpi_free( &R ); mbedtls_mpi_free( &T ); mbedtls_mpi_free( &A ); mbedtls_mpi_free( &RR ); return( ret ); } /* * Pseudo-primality test: small factors, then Miller-Rabin */ int mbedtls_mpi_is_prime_ext( const mbedtls_mpi *X, int rounds, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { int ret = MBEDTLS_ERR_ERROR_CORRUPTION_DETECTED; mbedtls_mpi XX; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( f_rng != NULL ); XX.s = 1; XX.n = X->n; XX.p = X->p; if( mbedtls_mpi_cmp_int( &XX, 0 ) == 0 || mbedtls_mpi_cmp_int( &XX, 1 ) == 0 ) return( MBEDTLS_ERR_MPI_NOT_ACCEPTABLE ); if( mbedtls_mpi_cmp_int( &XX, 2 ) == 0 ) return( 0 ); if( ( ret = mpi_check_small_factors( &XX ) ) != 0 ) { if( ret == 1 ) return( 0 ); return( ret ); } return( mpi_miller_rabin( &XX, rounds, f_rng, p_rng ) ); } #if !defined(MBEDTLS_DEPRECATED_REMOVED) /* * Pseudo-primality test, error probability 2^-80 */ int mbedtls_mpi_is_prime( const mbedtls_mpi *X, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( f_rng != NULL ); /* * In the past our key generation aimed for an error rate of at most * 2^-80. Since this function is deprecated, aim for the same certainty * here as well. */ return( mbedtls_mpi_is_prime_ext( X, 40, f_rng, p_rng ) ); } #endif /* * Prime number generation * * To generate an RSA key in a way recommended by FIPS 186-4, both primes must * be either 1024 bits or 1536 bits long, and flags must contain * MBEDTLS_MPI_GEN_PRIME_FLAG_LOW_ERR. */ int mbedtls_mpi_gen_prime( mbedtls_mpi *X, size_t nbits, int flags, int (*f_rng)(void *, unsigned char *, size_t), void *p_rng ) { #ifdef MBEDTLS_HAVE_INT64 // ceil(2^63.5) #define CEIL_MAXUINT_DIV_SQRT2 0xb504f333f9de6485ULL #else // ceil(2^31.5) #define CEIL_MAXUINT_DIV_SQRT2 0xb504f334U #endif int ret = MBEDTLS_ERR_MPI_NOT_ACCEPTABLE; size_t k, n; int rounds; mbedtls_mpi_uint r; mbedtls_mpi Y; MPI_VALIDATE_RET( X != NULL ); MPI_VALIDATE_RET( f_rng != NULL ); if( nbits < 3 || nbits > MBEDTLS_MPI_MAX_BITS ) return( MBEDTLS_ERR_MPI_BAD_INPUT_DATA ); mbedtls_mpi_init( &Y ); n = BITS_TO_LIMBS( nbits ); if( ( flags & MBEDTLS_MPI_GEN_PRIME_FLAG_LOW_ERR ) == 0 ) { /* * 2^-80 error probability, number of rounds chosen per HAC, table 4.4 */ rounds = ( ( nbits >= 1300 ) ? 2 : ( nbits >= 850 ) ? 3 : ( nbits >= 650 ) ? 4 : ( nbits >= 350 ) ? 8 : ( nbits >= 250 ) ? 12 : ( nbits >= 150 ) ? 18 : 27 ); } else { /* * 2^-100 error probability, number of rounds computed based on HAC, * fact 4.48 */ rounds = ( ( nbits >= 1450 ) ? 4 : ( nbits >= 1150 ) ? 5 : ( nbits >= 1000 ) ? 6 : ( nbits >= 850 ) ? 7 : ( nbits >= 750 ) ? 8 : ( nbits >= 500 ) ? 13 : ( nbits >= 250 ) ? 28 : ( nbits >= 150 ) ? 40 : 51 ); } while( 1 ) { MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( X, n * ciL, f_rng, p_rng ) ); /* make sure generated number is at least (nbits-1)+0.5 bits (FIPS 186-4 §B.3.3 steps 4.4, 5.5) */ if( X->p[n-1] < CEIL_MAXUINT_DIV_SQRT2 ) continue; k = n * biL; if( k > nbits ) MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( X, k - nbits ) ); X->p[0] |= 1; if( ( flags & MBEDTLS_MPI_GEN_PRIME_FLAG_DH ) == 0 ) { ret = mbedtls_mpi_is_prime_ext( X, rounds, f_rng, p_rng ); if( ret != MBEDTLS_ERR_MPI_NOT_ACCEPTABLE ) goto cleanup; } else { /* * An necessary condition for Y and X = 2Y + 1 to be prime * is X = 2 mod 3 (which is equivalent to Y = 2 mod 3). * Make sure it is satisfied, while keeping X = 3 mod 4 */ X->p[0] |= 2; MBEDTLS_MPI_CHK( mbedtls_mpi_mod_int( &r, X, 3 ) ); if( r == 0 ) MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( X, X, 8 ) ); else if( r == 1 ) MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( X, X, 4 ) ); /* Set Y = (X-1) / 2, which is X / 2 because X is odd */ MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &Y, X ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &Y, 1 ) ); while( 1 ) { /* * First, check small factors for X and Y * before doing Miller-Rabin on any of them */ if( ( ret = mpi_check_small_factors( X ) ) == 0 && ( ret = mpi_check_small_factors( &Y ) ) == 0 && ( ret = mpi_miller_rabin( X, rounds, f_rng, p_rng ) ) == 0 && ( ret = mpi_miller_rabin( &Y, rounds, f_rng, p_rng ) ) == 0 ) goto cleanup; if( ret != MBEDTLS_ERR_MPI_NOT_ACCEPTABLE ) goto cleanup; /* * Next candidates. We want to preserve Y = (X-1) / 2 and * Y = 1 mod 2 and Y = 2 mod 3 (eq X = 3 mod 4 and X = 2 mod 3) * so up Y by 6 and X by 12. */ MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( X, X, 12 ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_add_int( &Y, &Y, 6 ) ); } } } cleanup: mbedtls_mpi_free( &Y ); return( ret ); } #endif /* MBEDTLS_GENPRIME */ #if defined(MBEDTLS_SELF_TEST) #define GCD_PAIR_COUNT 3 static const int gcd_pairs[GCD_PAIR_COUNT][3] = { { 693, 609, 21 }, { 1764, 868, 28 }, { 768454923, 542167814, 1 } }; /* * Checkup routine */ int mbedtls_mpi_self_test( int verbose ) { int ret, i; mbedtls_mpi A, E, N, X, Y, U, V; mbedtls_mpi_init( &A ); mbedtls_mpi_init( &E ); mbedtls_mpi_init( &N ); mbedtls_mpi_init( &X ); mbedtls_mpi_init( &Y ); mbedtls_mpi_init( &U ); mbedtls_mpi_init( &V ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &A, 16, "EFE021C2645FD1DC586E69184AF4A31E" \ "D5F53E93B5F123FA41680867BA110131" \ "944FE7952E2517337780CB0DB80E61AA" \ "E7C8DDC6C5C6AADEB34EB38A2F40D5E6" ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &E, 16, "B2E7EFD37075B9F03FF989C7C5051C20" \ "34D2A323810251127E7BF8625A4F49A5" \ "F3E27F4DA8BD59C47D6DAABA4C8127BD" \ "5B5C25763222FEFCCFC38B832366C29E" ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &N, 16, "0066A198186C18C10B2F5ED9B522752A" \ "9830B69916E535C8F047518A889A43A5" \ "94B6BED27A168D31D4A52F88925AA8F5" ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &X, &A, &N ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &U, 16, "602AB7ECA597A3D6B56FF9829A5E8B85" \ "9E857EA95A03512E2BAE7391688D264A" \ "A5663B0341DB9CCFD2C4C5F421FEC814" \ "8001B72E848A38CAE1C65F78E56ABDEF" \ "E12D3C039B8A02D6BE593F0BBBDA56F1" \ "ECF677152EF804370C1A305CAF3B5BF1" \ "30879B56C61DE584A0F53A2447A51E" ) ); if( verbose != 0 ) mbedtls_printf( " MPI test #1 (mul_mpi): " ); if( mbedtls_mpi_cmp_mpi( &X, &U ) != 0 ) { if( verbose != 0 ) mbedtls_printf( "failed\n" ); ret = 1; goto cleanup; } if( verbose != 0 ) mbedtls_printf( "passed\n" ); MBEDTLS_MPI_CHK( mbedtls_mpi_div_mpi( &X, &Y, &A, &N ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &U, 16, "256567336059E52CAE22925474705F39A94" ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &V, 16, "6613F26162223DF488E9CD48CC132C7A" \ "0AC93C701B001B092E4E5B9F73BCD27B" \ "9EE50D0657C77F374E903CDFA4C642" ) ); if( verbose != 0 ) mbedtls_printf( " MPI test #2 (div_mpi): " ); if( mbedtls_mpi_cmp_mpi( &X, &U ) != 0 || mbedtls_mpi_cmp_mpi( &Y, &V ) != 0 ) { if( verbose != 0 ) mbedtls_printf( "failed\n" ); ret = 1; goto cleanup; } if( verbose != 0 ) mbedtls_printf( "passed\n" ); MBEDTLS_MPI_CHK( mbedtls_mpi_exp_mod( &X, &A, &E, &N, NULL ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &U, 16, "36E139AEA55215609D2816998ED020BB" \ "BD96C37890F65171D948E9BC7CBAA4D9" \ "325D24D6A3C12710F10A09FA08AB87" ) ); if( verbose != 0 ) mbedtls_printf( " MPI test #3 (exp_mod): " ); if( mbedtls_mpi_cmp_mpi( &X, &U ) != 0 ) { if( verbose != 0 ) mbedtls_printf( "failed\n" ); ret = 1; goto cleanup; } if( verbose != 0 ) mbedtls_printf( "passed\n" ); MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &X, &A, &N ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &U, 16, "003A0AAEDD7E784FC07D8F9EC6E3BFD5" \ "C3DBA76456363A10869622EAC2DD84EC" \ "C5B8A74DAC4D09E03B5E0BE779F2DF61" ) ); if( verbose != 0 ) mbedtls_printf( " MPI test #4 (inv_mod): " ); if( mbedtls_mpi_cmp_mpi( &X, &U ) != 0 ) { if( verbose != 0 ) mbedtls_printf( "failed\n" ); ret = 1; goto cleanup; } if( verbose != 0 ) mbedtls_printf( "passed\n" ); if( verbose != 0 ) mbedtls_printf( " MPI test #5 (simple gcd): " ); for( i = 0; i < GCD_PAIR_COUNT; i++ ) { MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &X, gcd_pairs[i][0] ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &Y, gcd_pairs[i][1] ) ); MBEDTLS_MPI_CHK( mbedtls_mpi_gcd( &A, &X, &Y ) ); if( mbedtls_mpi_cmp_int( &A, gcd_pairs[i][2] ) != 0 ) { if( verbose != 0 ) mbedtls_printf( "failed at %d\n", i ); ret = 1; goto cleanup; } } if( verbose != 0 ) mbedtls_printf( "passed\n" ); cleanup: if( ret != 0 && verbose != 0 ) mbedtls_printf( "Unexpected error, return code = %08X\n", (unsigned int) ret ); mbedtls_mpi_free( &A ); mbedtls_mpi_free( &E ); mbedtls_mpi_free( &N ); mbedtls_mpi_free( &X ); mbedtls_mpi_free( &Y ); mbedtls_mpi_free( &U ); mbedtls_mpi_free( &V ); if( verbose != 0 ) mbedtls_printf( "\n" ); return( ret ); } #endif /* MBEDTLS_SELF_TEST */ #endif /* MBEDTLS_BIGNUM_C */